Finite Group Databasebeta

← #276009

Subgroups — #276009

Order 512.

593 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33196C211yesmaximal, characteristic
2#28748C211yesmaximal, characteristic
3#22825C211yesmaximal, characteristic
4C4×#477C211yesmaximal, characteristic
5#10486C212yesmaximal
6#10277C212yesmaximal
7#9745C212yesmaximal
8#9692C211yesmaximal, characteristic
9#9602C211yesmaximal, characteristic
10#9569C212yesmaximal
11#8587C211yesmaximal, characteristic
12C2×#487C2212yes—
13C2×#477C2211yescharacteristic
14C2×#475C2212yes—
15#2868C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17C42×Q8C2211yescharacteristic
18#2043—22no—
19#1988C2212yes—
20#1987C2211yescharacteristic
21#1984C2212yes—
22#1981—21no—
23#1981—21no—
24#1956C2212yes—
25#1948—21no—
26#1948—21no—
27#1947C2211yescharacteristic
28#1921—21no—
29#1917—22no—
30#1863—21no—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1831—21no—
34#1803—21no—
35#1803—21no—
36#1803C2211yescharacteristic
37C4×#107—21no—
38#1770C2212yes—
39#1767—21no—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×#103—21no—
43C2×#334C2212yes—
44C2×#333C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8C2312yes—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4×Q8—21no—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57#487—24no—
58#487—42no—
59#477—22no—
60#477—22no—
61#475—42no—
62#475—42no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#106—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8D411yescharacteristic
71C2×C4⋊C8D411yescharacteristic
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×#103—21no—
77C2×#103—22no—
78C2×#103—21no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#97—22no—
84C2×C4×C8—21no—
85C2×C4×C8C2312yes—
86#391—21no—
87#389—21no—
88#389—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384C2311yescharacteristic
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4D411yescharacteristic
100C4×C4⋊C4D411yescharacteristic
101C4×C4⋊C4C2311yescharacteristic
102C43—21no—
103C43C2311yescharacteristic
104#340—22no—
105#340—22no—
106#340—21no—
107#340—21no—
108#340—21no—
109#340—21no—
110#336—22no—
111#336—22no—
112#336—22no—
113#336—21no—
114#336D412yes—
115#334—42no—
116#333—42no—
117#332—24no—
118#332—42no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8C2×D411yescharacteristic
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8C2×D412yes—
127C22×C8C2×D412yes—
128C4⋊Q8—21no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—42no—
136C4⋊Q8—42no—
137C4⋊Q8—81no—
138C4⋊Q8—81no—
139C4⋊Q8#4112yes—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×Q8—21no—
146C4×Q8—21no—
147C4×Q8—41no—
148C4×Q8—82no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—41no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42C2411yescharacteristic
181#107—24no—
182#107—42no—
183#106—44no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—21no—
187C4⋊C8—22no—
188C4⋊C8—21no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—21no—
192C4⋊C8—22no—
193C4⋊C8—42no—
194C4⋊C8—42no—
195C4⋊C8—42no—
196C4⋊C8—81no—
197C4⋊C8#4112yes—
198#103—24no—
199#103—44no—
200#103—44no—
201#103—44no—
202#103—42no—
203#103—82no—
204#103—82no—
205#97—82no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—42no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—41no—
213#95—41no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—22no—
218C2×Q8—21no—
219C2×Q8—21no—
220C2×Q8—41no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—81no—
228C2×Q8—81no—
229C2×Q8#13712yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C2×#4111yescharacteristic
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—42no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—81no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—22no—
295C42—41no—
296C42—42no—
297C42—41no—
298C42—41no—
299C42—41no—
300C42—41no—
301C42—42no—
302C42—42no—
303C42—81no—
304C42—82no—
305C42#13512yes—
306C42#13712yes—
307C42#13712yes—
308C42#13712yes—
309C42C2×#4111yescharacteristic
310C42C2×#4111yescharacteristic
311C42H(5,𝔽2)11yescharacteristic
312C42H(5,𝔽2)11yescharacteristic
313C23C2×#12111yescharacteristic
314Q8—22no—
315Q8—22no—
316Q8—41no—
317Q8—41no—
318Q8—82no—
319Q8—82no—
320Q8—82no—
321Q8—82no—
322Q8—82no—
323Q8—82no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4#46412yes—
355C2×C4#46412yes—
356C2×C4#47012yes—
357C2×C4#47012yes—
358C2×C4#47112yes—
359C2×C4#52511yescharacteristic
360C2×C4#53511yescharacteristic
361C2×C4#54611yescharacteristic
362C2×C4#55011yescharacteristic
363C2×C4#55311yescharacteristic
364C2×C4#55511yescharacteristic
365C2×C4C2×#13511yescharacteristic
366C2×C4C2×#13711yescharacteristic
367C2×C4C2×#13711yescharacteristic
368C2×C4C2×#13711yescharacteristic
369C2×C4C2×#13711yescharacteristic
370C2×C4#57511yescharacteristic
371C2×C4#57611yescharacteristic
372C2×C4#57711yescharacteristic
373C8—42no—
374C8—42no—
375C8—42no—
376C8—82no—
377C8—82no—
378C8—81no—
379C22#241411yescharacteristic
380C22C2×#46411yescharacteristic
381C22#304511yescharacteristic
382C22C2×#47011yescharacteristic
383C22#305611yescharacteristic
384C22C2×#47111yescharacteristic
385C22#306911yescharacteristic
386C4—21no—
387C4—21no—
388C4—22no—
389C4—22no—
390C4—41no—
391C4—42no—
392C4—81no—
393C4—81no—
394C4—82no—
395C4—81no—
396C4—161no—
397C4—162no—
398C4#170414yes—
399C4#311311yescharacteristic
400C4#311611yescharacteristic
401C4#321611yescharacteristic
402C4#321811yescharacteristic
403C4#330411yescharacteristic
404C4#333311yescharacteristic
405C4#339611yescharacteristic
406C4#340011yescharacteristic
407C2C2×#170411yescharacteristic
408C2#1917511yescharacteristic
409C2#1944711yescharacteristic
410C2#1949411yescharacteristic
411C2#2463311yescharacteristic
412C2#2466011yescharacteristic
413C2#2467111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#331962
#287482
#228252
C4×#4772
#104862
#102772
#97452
#96922
#96022
#95692
#85872